Sometimes the stems will root. Takes ages in the autumn, though. They might sprout in water and form a small tuber with roots but they take a while to become 'real' tubers. When you thin out the stems in spring, if you're growing for 'show', those ones tend to take much quicker.

I've planted up very late-bought tubers (cut price); they've emerged and leafed up, but the shorter daylight length brought them to dormancy no matter what I did. They 'know'!

A followup to my own post: the gorgeous, heavily blooming begonia I took in the house quickly fell apart, with branches dying for no apparent reason. It now looks terrible. Either it didn't like the transition to the heated house, or it knew it was "time", or somthing. I'm not sure if I should wait to cut off the branches until they all die or do it now.

The other ones (that I had potted myself) have all stopped blooming. No die back yet but new growth is leggy. I will dig and store them all but I'm not sure when I should do it: now or wait.